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Register a new userAdapting to Function Difficulty and Growth Conditions in Private Optimization
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Added by
Daniel Levy
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We develop algorithms for private stochastic convex optimization that adapt to the hardness of the specific function we wish to optimize. While previous work provide worst-case bounds for arbitrary convex functions, it is often the case that the function at hand belongs to a smaller class that enjoys faster rates. Concretely, we show that for functions exhibiting $kappa$-growth around the optimum, i.e., $f(x) ge f(x^*) + lambda kappa^{-1} |x-x^*|_2^kappa$ for $kappa > 1$, our algorithms improve upon the standard ${sqrt{d}}/{nvarepsilon}$ privacy rate to the faster $({sqrt{d}}/{nvarepsilon})^{tfrac{kappa}{kappa - 1}}$. Crucially, they achieve these rates without knowledge of the growth constant $kappa$ of the function. Our algorithms build upon the inverse sensitivity mechanism, which adapts to instance difficulty (Asi & Duchi, 2020), and recent localization techniques in private optimization (Feldman et al., 2020). We complement our algorithms with matching lower bounds for these function classes and demonstrate that our adaptive algorithm is emph{simultaneously} (minimax) optimal over all $kappa ge 1+c$ whenever $c = Theta(1)$.
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This paper presents the private-outsourced-Gaussian process-upper confidence bound (PO-GP-UCB) algorithm, which is the first algorithm for privacy-preserving Bayesian optimization (BO) in the outsourced setting with a provable performance guarantee. We consider the outsourced setting where the entity holding the dataset and the entity performing BO are represented by different parties, and the dataset cannot be released non-privately. For example, a hospital holds a dataset of sensitive medical records and outsources the BO task on this dataset to an industrial AI company. The key idea of our approach is to make the BO performance of our algorithm similar to that of non-private GP-UCB run using the original dataset, which is achieved by using a random projection-based transformation that preserves both privacy and the pairwise distances between inputs. Our main theoretical contribution is to show that a regret bound similar to that of the standard GP-UCB algorithm can be established for our PO-GP-UCB algorithm. We empirically evaluate the performance of our PO-GP-UCB algorithm with synthetic and real-world datasets.
In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. In this work, we present interactive shuffle protocols for stochastic convex optimization. Our optimization protocols rely on a new noninteractive protocol for summing vectors of bounded $ell_2$ norm. By combining this sum subroutine with techniques including mini-batch stochastic gradient descent, accelerated gradient descent, and Nesterovs smoothing method, we obtain loss guarantees for a variety of convex loss functions that significantly improve on those of the local model and sometimes match those of the central model.
Through the lens of information-theoretic reductions, we examine a reductions approach to fair optimization and learning where a black-box optimizer is used to learn a fair model for classification or regression. Quantifying the complexity, both statistically and computationally, of making such models satisfy the rigorous definition of differential privacy is our end goal. We resolve a few open questions and show applicability to fair machine learning, hypothesis testing, and to optimizing non-standard measures of classification loss. Furthermore, our sample complexity bounds are tight amongst all strategies that jointly minimize a composition of functions. The reductions approach to fair optimization can be abstracted as the constrained group-objective optimization problem where we aim to optimize an objective that is a function of losses of individual groups, subject to some constraints. We give the first polynomial-time algorithms to solve the problem with $(epsilon, 0)$ or $(epsilon, delta)$ differential privacy guarantees when defined on a convex decision set (for example, the $ell_P$ unit ball) with convex constraints and losses. Accompanying information-theoretic lower bounds for the problem are presented. In addition, compared to a previous method for ensuring differential privacy subject to a relaxed form of the equalized odds fairness constraint, the $(epsilon, delta)$-differentially private algorithm we present provides asymptotically better sample complexity guarantees, resulting in an exponential improvement in certain parameter regimes. We introduce a class of bounded divergence linear optimizers, which could be of independent interest, and specialize to pure and approximate differential privacy.
Finding efficient, easily implementable differentially private (DP) algorithms that offer strong excess risk bounds is an important problem in modern machine learning. To date, most work has focused on private empirical risk minimization (ERM) or private population loss minimization. However, there are often other objectives--such as fairness, adversarial robustness, or sensitivity to outliers--besides average performance that are not captured in the classical ERM setup. To this end, we study a completely general family of convex, Lipschitz loss functions and establish the first known DP excess risk and runtime bounds for optimizing this broad class. We provide similar bounds under additional assumptions of smoothness and/or strong convexity. We also address private stochastic convex optimization (SCO). While $(epsilon, delta)$-DP ($delta > 0$) has been the focus of much recent work in private SCO, proving tight population loss bounds and runtime bounds for $(epsilon, 0)$-DP remains a challenging open problem. We provide the tightest known $(epsilon, 0)$-DP population loss bounds and fastest runtimes under the presence of (or lack of) smoothness and strong convexity. Our methods extend to the $delta > 0$ setting, where we offer the unique benefit of ensuring differential privacy for arbitrary $epsilon > 0$ by incorporating a new form of Gaussian noise. Finally, we apply our theory to two learning frameworks: tilted ERM and adversarial learning. In particular, our theory quantifies tradeoffs between adversarial robustness, privacy, and runtime. Our results are achieved using perhaps the simplest DP algorithm: output perturbation. Although this method is not novel conceptually, our novel implementation scheme and analysis show that the power of this method to achieve strong privacy, utility, and runtime guarantees has not been fully appreciated in prior works.
We study differentially private (DP) algorithms for stochastic convex optimization (SCO). In this problem the goal is to approximately minimize the population loss given i.i.d. samples from a distribution over convex and Lipschitz loss functions. A long line of existing work on private convex optimization focuses on the empirical loss and derives asymptotically tight bounds on the excess empirical loss. However a significant gap exists in the known bounds for the population loss. We show that, up to logarithmic factors, the optimal excess population loss for DP algorithms is equal to the larger of the optimal non-private excess population loss, and the optimal excess empirical loss of DP algorithms. This implies that, contrary to intuition based on private ERM, private SCO has asymptotically the same rate of $1/sqrt{n}$ as non-private SCO in the parameter regime most common in practice. The best previous result in this setting gives rate of $1/n^{1/4}$. Our approach builds on existing differentially private algorithms and relies on the analysis of algorithmic stability to ensure generalization.
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